Question

a guidance counselor wants to determine if there is a relationship between a students number of absences, ( x ), and their grade point average (gpa), ( y ). the data that were collected are displayed in the scatterplot and the least - squares regression line was calculated. one student with 2 absences has a gpa of 1.8. this point is circled on the graph. what effect does the circled point have on the standard deviation of the residuals? this point will increase the value of the standard deviation of the residuals because it has a large positive residual. this point will increase the value of the standard deviation of the residuals because it has a large negative residual. this point will not affect the value of the standard deviation of the residuals because it has a large positive residual. this point will decrease the value of the standard deviation of the residuals because it has a large negative residual.

Explanation:

Step1: Recall the formula for residual

Residual \(e=y - \hat{y}\), where \(y\) is the observed value and \(\hat{y}\) is the predicted value.

Step2: Analyze the position of the circled point

Looking at the scatter - plot and the regression line, for the student with \(x = 2\) absences, the observed \(y=1.8\) (GPA) is much lower than the predicted \(\hat{y}\) (since the regression line is above the point at \(x = 2\)). So, \(e=y-\hat{y}<0\) (negative residual).

Step3: Recall the formula for standard deviation of residuals

The formula for the standard deviation of residuals \(s_{e}=\sqrt{\frac{\sum_{i = 1}^{n}(y_{i}-\hat{y}_{i})^{2}}{n - 2}}\). A large value of \(|y_{i}-\hat{y}_{i}|\) (in this case, a large negative value, so \(|e|\) is large) will increase the value of \(\sum_{i = 1}^{n}(y_{i}-\hat{y}_{i})^{2}\), and thus increase \(s_{e}\)

Answer:

This point will increase the value of the standard deviation of the residuals because it has a large negative residual.